I like the historical relevance of your proposal, including the relationship of the Kennedy brothers, especially after the Cuban Missile Crisis. We can think of other "hot moments" where there might be drive to push the project further.
I also agree that the time travel angle is problematic, as are many of the psychic issues.
One alternative might be something like this.
A small group of industrialists meet following the detonation of the Hiroshima and Nagasaki bombs. These industrialists had been asked to provide financial and technical support to the Manhattan Project but in the process had also established an applied sciences team that had relied on statistical analysis of the likely consequences of the bombing. Statistical probability theory had allowed these scientists to assemble a great array of data and compute models with a high degree of statistical certainty of future events. While much of the country celebrates the end of the war, these industrialists are already calculating the likely long-term consequences, and the probability of total nuclear warfare.
This group becomes the core of the Council of Tomorrow. The head of the applied sciences/forecasting group was Morrow, a mathematical genius. Much later this same group would help establish much of the predictive theory behind the Club of Rome's Limits to Growth thesis.
http://en.wikipedia.org/wiki/The_Limits_to_Growth
Better modeling and statistical methods allowed for greater predictive accuracy, but never perfect prediction. But under Morrow, a broader array of potential threats- both man-made and natural- were factored into the study, suggesting an inevitable catastrophe, but of uncertain nature, sometime before 2050. While the Council had anticipated the Cold War was likely to become hot before 1992, following the end of the Cold War, the scholars focused on the likelihood of environmental crisis triggering wars among states, with similar consequences, with growing certainty as time progressed.
This approach would basically allow you to incorporate other types of catastrophe or could even make the "end of the world" one of the mysteries your Morrow Team has to learn. Perhaps they even carry the seed of a future catastrophe in their ranks.